Stability of relaxed calibration

Abstract

Estimation of the population total of a variable can be improved by calibration on a set of auxiliary variables. It is difficult to establish that such a set of variables is sufficient, that estimation could not be improved by calibration on any further variables. We address this issue by finding an upper bound for the change of the calibration estimate of the population total of a variable when the auxiliary information is supplemented by another variable for which the population total is known. This upper bound can be interpreted as a measure of sensitivity of the estimate to unavailable auxiliary information and considered as a factor in deciding whether to seek further data sources that would be included in calibration.

Figures (12)

• Figure 1: The simulated estimates $\Delta{\hat{\theta}}$ and $\Delta{\hat{\theta}}^{(0)}$ (left) and their ratios (right). The line drawn by dashes on the left indicates the identity.
• Figure 2: Histograms of the sampling probabilities (left) and outcomes (right) in population ${\cal{P}}$. The mean is marked by solid vertical line and median by dashes.
• Figure 3: Solution of the equation $\hbox{\bf x}_{K+1}^{\top}\hbox{\bf x}_{K+1} = n$.
• Figure 4: Estimates $\Delta{\hat{\theta}}$ and $\Delta{\hat{\theta}}^{(0)}$ (left) and their ratios (right). The dashes in the left-hand plot mark the identity line.
• Figure 5: $\Delta{\hat{\theta}}$ and $\Delta{\hat{\theta}}^{(0)}$ as functions of $t_{K+1\,}$. The values are drawn for the constraints $t_{K+1} \le T$ by solid lines and $t_{K+1} = T$ by dashes.